Group #38742831 — subgroups

Subgroups — #38742831

Order 25216.

216 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#13096802	C₂	1	1	yes	maximal, characteristic
2	#27160925	C₂	1	1	yes	maximal, characteristic
3	#27160833	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13096802	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
7	C₄⋊#13096802	—	2	2	no	—
8	#20263233	C₄	1	2	yes	—
9	#20263233	C₄	1	2	yes	—
10	C₂²×C₈×C₁₉₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₉₇	—	2	2	no	—
13	C₂²×Dic₁₉₇	C₂×C₄	1	4	yes	—
14	C₂×#13096802	—	2	2	no	—
15	C₂×#13096802	—	2	4	no	—
16	C₄×Dic₁₉₇	—	2	2	no	—
17	C₄×Dic₁₉₇	—	2	2	no	—
18	C₂²×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
21	C₂×C₈×C₁₉₇	—	2	2	no	—
22	C₂×C₈×C₁₉₇	—	2	4	no	—
23	C₄²×C₁₉₇	D₄	1	2	yes	—
24	C₄²×C₁₉₇	Q₈	1	2	yes	—
25	C₂×Dic₁₉₇	—	2	4	no	—
26	C₂×Dic₁₉₇	—	2	4	no	—
27	C₂×Dic₁₉₇	—	2	4	no	—
28	#13096802	—	4	2	no	—
29	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₉₇	—	2	2	no	—
31	C₂×C₄×C₁₉₇	—	2	2	no	—
32	C₂×C₄×C₁₉₇	C₄⋊C₄	1	2	yes	—
33	C₂×C₄×C₁₉₇	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₁₉₇	#41	1	2	yes	—
37	C₂×C₄×C₁₉₇	#41	1	2	yes	—
38	C₈×C₁₉₇	—	4	2	no	—
39	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₁₉₇	#117	1	2	yes	—
41	C₂²×C₁₉₇	#131	1	2	yes	—
42	C₂²×C₁₉₇	#131	1	2	yes	—
43	C₄×C₁₉₇	—	2	2	no	—
44	C₄×C₁₉₇	—	2	2	no	—
45	C₄×C₁₉₇	#124	1	1	yes	characteristic
46	C₄×C₁₉₇	#124	1	1	yes	characteristic
47	C₄×C₁₉₇	#125	1	2	yes	—
48	Dic₁₉₇	—	4	4	no	—
49	C₂×C₁₉₇	#382	1	1	yes	characteristic
50	C₂×C₁₉₇	#424	1	1	yes	characteristic
51	C₂×C₁₉₇	#424	1	1	yes	characteristic
52	C₂×C₁₉₇	#433	1	4	yes	—
53	C₁₉₇	#1832	1	1	yes	characteristic, complemented, Sylow
54	#1832	—	197	1	no	maximal, Sylow
55	C₂×C₄⋊C₈	—	197	1	no	—
56	C₂×C₄⋊C₈	D₁₉₇	1	1	yes	characteristic
57	#376	—	197	1	no	—
58	#336	—	197	4	no	—
59	C₂²×C₈	—	197	2	no	—
60	C₂²×C₈	C₂×D₁₉₇	1	2	yes	—
61	C₂×C₄²	—	197	2	no	—
62	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	—	394	2	no	—
65	#95	—	197	2	no	—
66	#95	—	197	2	no	—
67	C₂²×C₄	—	197	4	no	—
68	C₂²×C₄	C₄×D₁₉₇	1	1	yes	characteristic
69	C₂²×C₄	C₄×D₁₉₇	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	394	2	no	—
74	C₂×C₈	—	394	4	no	—
75	C₄²	—	394	2	no	—
76	C₄²	—	394	2	no	—
77	C₄²	C₂²⋊D₁₉₇	1	2	yes	—
78	C₄²	Dic₃₉₄	1	2	yes	—
79	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	394	4	no	—
83	C₂×C₄	—	394	4	no	—
84	C₂×C₄	—	394	4	no	—
85	C₂×C₄	#15691337	1	2	yes	—
86	C₂×C₄	#15691337	1	2	yes	—
87	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₃₉₄	1	1	yes	characteristic
89	C₂×C₄	#15691357	1	2	yes	—
90	C₂×C₄	#15691359	1	2	yes	—
91	C₈	—	4	2	no	—
92	C₈	—	788	2	no	—
93	C₂²	C₂×#15691337	1	1	yes	characteristic
94	C₂²	#20263284	1	2	yes	—
95	C₂²	#20263332	1	2	yes	—
96	C₂²	#20263334	1	2	yes	—
97	C₄	—	2	2	no	—
98	C₄	—	2	2	no	—
99	C₄	—	788	4	no	—
100	C₄	#20263313	1	1	yes	characteristic
101	C₄	#20263315	1	1	yes	characteristic
102	C₄	#20263317	1	2	yes	—
103	C₂	#27160944	1	1	yes	characteristic
104	C₂	#27161105	1	1	yes	characteristic
105	C₂	#27161107	1	1	yes	characteristic
106	C₂	#27161139	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#13096802

C₂

yes

maximal, characteristic

#27160925

C₂

yes

maximal, characteristic

#27160833

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×#13096802

C₂²

yes

—

C₂×C₄×Dic₁₉₇

C₂²

yes

—

C₄⋊#13096802

—

#20263233

C₄

yes

—

#20263233

C₄

yes

—

C₂²×C₈×C₁₉₇

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₇

—

C₂²×Dic₁₉₇

C₂×C₄

yes

—

C₂×#13096802

—

C₂×#13096802

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₄²×C₁₉₇

D₄

yes

—

C₄²×C₁₉₇

Q₈

yes

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

C₂³×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₂×C₄×C₁₉₇

#41

yes

—

C₈×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₇

#117

yes

—

C₂²×C₁₉₇

#131

yes

—

C₂²×C₁₉₇

#131

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

yes

C₄×C₁₉₇

yes

C₄×C₁₉₇

yes

—

Dic₁₉₇

—

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

—

C₁₉₇

#1832

yes

characteristic, complemented, Sylow

#1832

—

197

maximal, Sylow

C₂×C₄⋊C₈

—

197

—

C₂×C₄⋊C₈

D₁₉₇

yes

characteristic

#376

—

197

—

#336

—

197

—

C₂²×C₈

—

197

—

C₂²×C₈

C₂×D₁₉₇

yes

—

C₂×C₄²

—

197

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

394

—

#95

—

197

—

#95

—

197

—

C₂²×C₄

—

197

—

C₂²×C₄

C₄×D₁₉₇

yes

characteristic

C₂²×C₄

C₄×D₁₉₇

yes

characteristic

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₄²

—

394

—

C₄²

—

394

—

C₄²

C₂²⋊D₁₉₇

yes

—

C₄²

Dic₃₉₄

yes

—

C₂³

C₂×C₄×D₁₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

394

—

C₂×C₄

—

394

—

C₂×C₄

—

394

—

C₂×C₄

#15691337

yes

—

C₂×C₄

#15691337

yes

—

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

C₂×C₄

C₂×Dic₃₉₄

yes

characteristic

C₂×C₄

#15691357

yes

—

C₂×C₄

#15691359

yes

—

C₈

—

C₈

—

788

—

C₂²

C₂×#15691337

yes

characteristic

C₂²

#20263284

yes

—

C₂²

#20263332

yes

—

C₂²

#20263334

yes

—

C₄

—

C₄

—

C₄

—

788

—

100

C₄

#20263313

yes

characteristic

101

C₄

#20263315

yes

characteristic

102

C₄

#20263317

yes

—

103

C₂

#27160944

yes

characteristic

104

C₂

#27161105

yes

characteristic

105

C₂

#27161107

yes

characteristic

106

C₂

#27161139

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₉₇
commutator, socle, prime radical	C₂³×C₁₉₇	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₁₉₇	C₂
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₉₇
hypocenter	C₁₉₇	#1832
LCS	G ▷ C₂³×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₉₇

commutator, socle, prime radical

C₂³×C₁₉₇

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₁₉₇

C₂

cosocle

C₂²×C₄×C₁₉₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₉₇

C₁₉₇

G ▷ C₂³×C₁₉₇ ▷ C₁₉₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
53. C₁₉₇	#1832	semidirect

H ≅ G/N

type

53. C₁₉₇

#1832

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄⋊#13096802	2
#27160925	2
#27160833	2
C₂×C₄⋊C₈×C₁₉₇	2
#1832	197

subgroup

index

C₂×C₄⋊C₈×C₁₉₇

197

Subgroups — #38742831

Conjugacy classes of subgroups

Complements

Maximal subgroups